Physlib.FluidDynamics.CauchyFlow.Newtonian

A Cauchy flow in $d$-dimensional space is Newtonian with respect to the scalar fields of pressure $p$, shear viscosity $\mu$, and second viscosity $\lambda$ if its stress tensor $\mathbf{\sigma}(t, \mathbf{x})$ satisfies the Newtonian constitutive relation: $$\mathbf{\sigma}(t, \mathbf{x}) = -p(t, \mathbf{x}) \mathbf{I} + \mu(t, \mathbf{x}) \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^\intercal \right) + \lambda(t, \mathbf{x}) (\nabla \cdot \mathbf{u}) \mathbf{I}$$ for all times $t$ and spatial positions $\mathbf{x}$, where $\mathbf{u}$ is the velocity field of the flow, $\mathbf{I}$ is the $d \times d$ identity matrix, $\nabla \mathbf{u}$ is the spatial velocity gradient, and $\nabla \cdot \mathbf{u}$ is the divergence of the velocity field.